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Statement

Théorème 2 (p. 124). Let $A$ be a highly composite number and $p$ its largest prime factor. Let $\lambda<p$ be a prime and $b=v_\lambda(A)$, the exponent of $\lambda$ in $A$. Then

$$\log\Bigl(1+\frac1b\Bigr)\ \ge\ \frac{\log\lambda\,\log2}{\log p}+O(p^{-\gamma}), \qquad \log\Bigl(1+\frac1{b+1}\Bigr)\ \le\ \frac{\log\lambda\,\log2}{\log p}+O(p^{-\gamma}).$$

Here $\gamma>0$ is the constant of Théorème 1. The paper says (p. 125) that the theorem improves Alaoglu and Erdős's Theorems 11 and 12 (Trans. Amer. Math. Soc. 56 (1944)), and that the remark before it (p. 124) would allow a smaller error than $O(p^{-\gamma})$ when $b$ is small.

Proof pointer

Pp. 123–125. Proposition 6 (p. 123), a consequence of Théorème 1, shows that $v_\lambda(A)$ can differ from the exponent of $\lambda$ in the preceding superior highly composite number $N_\epsilon$ only for $\lambda$ within $Cx^{-\gamma}$ of a threshold in the scale $\epsilon\log\lambda$, giving $\log(1+1/(b+1))-Cx^{-\gamma}\le\epsilon\log\lambda\le\log(1+1/b)+Cx^{-\gamma}$. The corollary of Proposition 4 gives $p-x=O(x^\tau)$, so $p\sim x$, and replacing $\epsilon=\log2/\log x$ by $\log2/\log p$ costs less than $O(p^{-\gamma})$ because $\gamma<1-\tau$ (p. 125).

Dependencies

Théorème 1; the paper's Propositions 4 and 6 and the corollary of Proposition 4 (pp. 120, 123); Ingham's theorem on primes in short intervals (display (4), p. 116).

Read depth

Claims checked: the statement and the remarks around it were read clause by clause on the page images of the print. The proof was read for its structure and is not reconstructed or independently reviewed here.

Source. Jean-Louis Nicolas, Répartition des nombres hautement composés de Ramanujan, Canadian J. Math. 23 (1971), no. 1, 116–130, doi:10.4153/cjm-1971-012-6; the edition read is named on the source card.

Bears on

No Erdős problem directly; the theorem describes the exponents of highly composite numbers and is not used in the paper's counting results.